Graphics.Rendering.Plot.Render.Plot.Axis:renderAxisTick from plot-0.2.3.4, B

Average Error: 16.3 → 9.6

Time: 9.1s

Precision: 64

Math

\[\left(x + y\right) - \frac{\left(z - t\right) \cdot y}{a - t}\]

↓

\[\begin{array}{l} \mathbf{if}\;t \le -1.028223383246293222355684506746970017299 \cdot 10^{170} \lor \neg \left(t \le 1.87872673246061357747397635459851376059 \cdot 10^{72} \lor \neg \left(t \le 5.370291603550818392691273515642313131572 \cdot 10^{292}\right)\right):\\ \;\;\;\;\frac{z \cdot y}{t} + x\\ \mathbf{else}:\\ \;\;\;\;x + \left(y - \frac{\sqrt[3]{z - t}}{\sqrt[3]{\sqrt[3]{a - t} \cdot \sqrt[3]{a - t}}} \cdot \left(\frac{\sqrt[3]{z - t}}{\sqrt[3]{\sqrt[3]{a - t}}} \cdot \left(\frac{\sqrt[3]{z - t}}{\sqrt[3]{a - t}} \cdot \frac{y}{\sqrt[3]{a - t}}\right)\right)\right)\\ \end{array}\]

C

double f(double x, double y, double z, double t, double a) {
        double r604438 = x;
        double r604439 = y;
        double r604440 = r604438 + r604439;
        double r604441 = z;
        double r604442 = t;
        double r604443 = r604441 - r604442;
        double r604444 = r604443 * r604439;
        double r604445 = a;
        double r604446 = r604445 - r604442;
        double r604447 = r604444 / r604446;
        double r604448 = r604440 - r604447;
        return r604448;
}

↓

double f(double x, double y, double z, double t, double a) {
        double r604449 = t;
        double r604450 = -1.0282233832462932e+170;
        bool r604451 = r604449 <= r604450;
        double r604452 = 1.8787267324606136e+72;
        bool r604453 = r604449 <= r604452;
        double r604454 = 5.370291603550818e+292;
        bool r604455 = r604449 <= r604454;
        double r604456 = !r604455;
        bool r604457 = r604453 || r604456;
        double r604458 = !r604457;
        bool r604459 = r604451 || r604458;
        double r604460 = z;
        double r604461 = y;
        double r604462 = r604460 * r604461;
        double r604463 = r604462 / r604449;
        double r604464 = x;
        double r604465 = r604463 + r604464;
        double r604466 = r604460 - r604449;
        double r604467 = cbrt(r604466);
        double r604468 = a;
        double r604469 = r604468 - r604449;
        double r604470 = cbrt(r604469);
        double r604471 = r604470 * r604470;
        double r604472 = cbrt(r604471);
        double r604473 = r604467 / r604472;
        double r604474 = cbrt(r604470);
        double r604475 = r604467 / r604474;
        double r604476 = r604467 / r604470;
        double r604477 = r604461 / r604470;
        double r604478 = r604476 * r604477;
        double r604479 = r604475 * r604478;
        double r604480 = r604473 * r604479;
        double r604481 = r604461 - r604480;
        double r604482 = r604464 + r604481;
        double r604483 = r604459 ? r604465 : r604482;
        return r604483;
}

Error

Bits error versus x

Bits error versus y

Bits error versus z

Bits error versus t

Bits error versus a

Target

Original	16.3
Target	8.4
Herbie	9.6

\[\begin{array}{l} \mathbf{if}\;\left(x + y\right) - \frac{\left(z - t\right) \cdot y}{a - t} \lt -1.366497088939072697550672266103566343531 \cdot 10^{-7}:\\ \;\;\;\;\left(y + x\right) - \left(\left(z - t\right) \cdot \frac{1}{a - t}\right) \cdot y\\ \mathbf{elif}\;\left(x + y\right) - \frac{\left(z - t\right) \cdot y}{a - t} \lt 1.475429344457723334351036314450840066235 \cdot 10^{-239}:\\ \;\;\;\;\frac{y \cdot \left(a - z\right) - x \cdot t}{a - t}\\ \mathbf{else}:\\ \;\;\;\;\left(y + x\right) - \left(\left(z - t\right) \cdot \frac{1}{a - t}\right) \cdot y\\ \end{array}\]

Derivation

1. Split input into 2 regimes

2. if t < -1.0282233832462932e+170 or 1.8787267324606136e+72 < t < 5.370291603550818e+292

   1. Initial program 30.4

      \[\left(x + y\right) - \frac{\left(z - t\right) \cdot y}{a - t}\]

   2. Taylor expanded around inf 17.4

      \[\leadsto \color{blue}{\frac{z \cdot y}{t} + x}\]

   if -1.0282233832462932e+170 < t < 1.8787267324606136e+72 or 5.370291603550818e+292 < t

   1. Initial program 10.0

      \[\left(x + y\right) - \frac{\left(z - t\right) \cdot y}{a - t}\]
   2. Using strategy rm

   3. Applied add-cube-cbrt 10.2

      \[\leadsto \left(x + y\right) - \frac{\left(z - t\right) \cdot y}{\color{blue}{\left(\sqrt[3]{a - t} \cdot \sqrt[3]{a - t}\right) \cdot \sqrt[3]{a - t}}}\]

   4. Applied times-frac 7.6

      \[\leadsto \left(x + y\right) - \color{blue}{\frac{z - t}{\sqrt[3]{a - t} \cdot \sqrt[3]{a - t}} \cdot \frac{y}{\sqrt[3]{a - t}}}\]
   5. Using strategy rm

   6. Applied add-cube-cbrt 7.7

      \[\leadsto \left(x + y\right) - \frac{\color{blue}{\left(\sqrt[3]{z - t} \cdot \sqrt[3]{z - t}\right) \cdot \sqrt[3]{z - t}}}{\sqrt[3]{a - t} \cdot \sqrt[3]{a - t}} \cdot \frac{y}{\sqrt[3]{a - t}}\]

   7. Applied times-frac 7.7

      \[\leadsto \left(x + y\right) - \color{blue}{\left(\frac{\sqrt[3]{z - t} \cdot \sqrt[3]{z - t}}{\sqrt[3]{a - t}} \cdot \frac{\sqrt[3]{z - t}}{\sqrt[3]{a - t}}\right)} \cdot \frac{y}{\sqrt[3]{a - t}}\]

   8. Applied associate-*l* 6.7

      \[\leadsto \left(x + y\right) - \color{blue}{\frac{\sqrt[3]{z - t} \cdot \sqrt[3]{z - t}}{\sqrt[3]{a - t}} \cdot \left(\frac{\sqrt[3]{z - t}}{\sqrt[3]{a - t}} \cdot \frac{y}{\sqrt[3]{a - t}}\right)}\]
   9. Using strategy rm

   10. Applied associate--l+ 5.3

       \[\leadsto \color{blue}{x + \left(y - \frac{\sqrt[3]{z - t} \cdot \sqrt[3]{z - t}}{\sqrt[3]{a - t}} \cdot \left(\frac{\sqrt[3]{z - t}}{\sqrt[3]{a - t}} \cdot \frac{y}{\sqrt[3]{a - t}}\right)\right)}\]
   11. Using strategy rm

   12. Applied add-cube-cbrt 5.7

       \[\leadsto x + \left(y - \frac{\sqrt[3]{z - t} \cdot \sqrt[3]{z - t}}{\sqrt[3]{\color{blue}{\left(\sqrt[3]{a - t} \cdot \sqrt[3]{a - t}\right) \cdot \sqrt[3]{a - t}}}} \cdot \left(\frac{\sqrt[3]{z - t}}{\sqrt[3]{a - t}} \cdot \frac{y}{\sqrt[3]{a - t}}\right)\right)\]

   13. Applied cbrt-prod 6.1

       \[\leadsto x + \left(y - \frac{\sqrt[3]{z - t} \cdot \sqrt[3]{z - t}}{\color{blue}{\sqrt[3]{\sqrt[3]{a - t} \cdot \sqrt[3]{a - t}} \cdot \sqrt[3]{\sqrt[3]{a - t}}}} \cdot \left(\frac{\sqrt[3]{z - t}}{\sqrt[3]{a - t}} \cdot \frac{y}{\sqrt[3]{a - t}}\right)\right)\]

   14. Applied times-frac 6.0

       \[\leadsto x + \left(y - \color{blue}{\left(\frac{\sqrt[3]{z - t}}{\sqrt[3]{\sqrt[3]{a - t} \cdot \sqrt[3]{a - t}}} \cdot \frac{\sqrt[3]{z - t}}{\sqrt[3]{\sqrt[3]{a - t}}}\right)} \cdot \left(\frac{\sqrt[3]{z - t}}{\sqrt[3]{a - t}} \cdot \frac{y}{\sqrt[3]{a - t}}\right)\right)\]

   15. Applied associate-*l* 6.2

       \[\leadsto x + \left(y - \color{blue}{\frac{\sqrt[3]{z - t}}{\sqrt[3]{\sqrt[3]{a - t} \cdot \sqrt[3]{a - t}}} \cdot \left(\frac{\sqrt[3]{z - t}}{\sqrt[3]{\sqrt[3]{a - t}}} \cdot \left(\frac{\sqrt[3]{z - t}}{\sqrt[3]{a - t}} \cdot \frac{y}{\sqrt[3]{a - t}}\right)\right)}\right)\]
3. Recombined 2 regimes into one program.

4. Final simplification 9.6

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \le -1.028223383246293222355684506746970017299 \cdot 10^{170} \lor \neg \left(t \le 1.87872673246061357747397635459851376059 \cdot 10^{72} \lor \neg \left(t \le 5.370291603550818392691273515642313131572 \cdot 10^{292}\right)\right):\\ \;\;\;\;\frac{z \cdot y}{t} + x\\ \mathbf{else}:\\ \;\;\;\;x + \left(y - \frac{\sqrt[3]{z - t}}{\sqrt[3]{\sqrt[3]{a - t} \cdot \sqrt[3]{a - t}}} \cdot \left(\frac{\sqrt[3]{z - t}}{\sqrt[3]{\sqrt[3]{a - t}}} \cdot \left(\frac{\sqrt[3]{z - t}}{\sqrt[3]{a - t}} \cdot \frac{y}{\sqrt[3]{a - t}}\right)\right)\right)\\ \end{array}\]

Reproduce

herbie shell --seed 2019356 
(FPCore (x y z t a)
  :name "Graphics.Rendering.Plot.Render.Plot.Axis:renderAxisTick from plot-0.2.3.4, B"
  :precision binary64

  :herbie-target
  (if (< (- (+ x y) (/ (* (- z t) y) (- a t))) -1.3664970889390727e-07) (- (+ y x) (* (* (- z t) (/ 1 (- a t))) y)) (if (< (- (+ x y) (/ (* (- z t) y) (- a t))) 1.4754293444577233e-239) (/ (- (* y (- a z)) (* x t)) (- a t)) (- (+ y x) (* (* (- z t) (/ 1 (- a t))) y))))

  (- (+ x y) (/ (* (- z t) y) (- a t))))